The 60D will be just fine for sports photography. I use a pair of them, but 7D would be only a mild upgrade for most people. Shooting with 50Ds in the past (same AF system as 60D) I could nail focus on moving subjects about 95% of the time. With 7D, it's 97% of the time. Is that level of improvement worth the upgrade? 8 fps vs 5 fps might get me a few more "perfect poses"... but also means more editing and trashed shots. I use 7D for their durability and sealing, mainly, as well as that last little edge in focus speed and accuracry. But in the right hands, 60D can do 90 or 95% of what 7D can do, and is a bit lighter, smaller too... plus has the nice articulated LCD, locking mode dial, which 7D doesn't (the dial lock can be added for about $100).
The 70-200/4 IS will not improve on image quality of your 55-250 significantly. However, it will focus faster and more accurately. It is also considerably better built and more durable lens, as well as better sealed against dust and moisture. The IS is more sophisticated, too. Might be a little more effective, but mainly it knows when to turn itself off, such as when locked down on a tripod. Also, you can fit it with a tripod mounting ring, which can be useful.
Your 60D is identical to 7D low light/high ISO performance. You should be able to shoot at ISO 3200 without much extra effort. Your best investment might be in a good noise reduction s'ware and add'l post processing skills. With some effort, you can shoot higher than 3200 with these cameras.
Yes, an f2.8 lens would be nice, but is bigger, heavier and more expensive... some skillful post processing might substitute. Rather than an f2.8 zoom, you might consider adding a few faster primes, which can be one or two stops faster than any zoom. It sort of depends upon what access you have to the events. If shooting from the sidelines, the focal lengths you need will be different from shooting from the bleachers.
If you are going to shoot commercially at the games (i.e., if you are going to get paid), you need permission from the school/organizer. Before starting to count on making any money, find out if there's already a photographer working the games. If the school/organizer has someone under contract, you might find yourself in violation of that agreement.
